Restriction categories were devised to study partial phenomena in an axiomatic setting. Here we give a whirlwind tour, more details can be found in [1]}, [2]}, [3]}. In a restriction category, every arrow \(f : A \rightarrow B\)  has an associated idempotent \(\overline{f} : A \rightarrow A\) , thought of as the identity function restricted to the domain of definition of \(f\) . We call them domain idempotents.
Arrows where \(\overline{f} = 1_A\)  are called total, and form a subcategory. Further, we have:
